Autocorrelation (words)

In combinatorics, a branch of mathematics, the autocorrelation of a word is the set of periods of this word. More precisely, it is a sequence of values which indicate how much the end of a word looks likes the beginning of a word. This value can be used to compute, for example, the average value of the first occurrence of this word in a random string.

Definition

In this article, A is an alphabet, and a word on A of length n. The autocorrelation of can be defined as the correlation of with itself. However, we redefine this notion below.

Autocorrelation vector

The autocorrelation vector of is , with being 1 if the prefix of length equals the suffix of length , and with being 0 otherwise. That is indicates whether .

For example, the autocorrelation vector of is since, clearly, for being 0, 1 or 2, the prefix of length is equal to the suffix of length . The autocorrelation vector of is since no strict prefix is equal to a strict suffix. Finally, the autocorrelation vector of is 100011, as shown in the following table:

a a b b a a
a a b b a a 1
a a b b a a 0
a a b b a a 0
a a b b a a 0
a a b b a a 1
a a b b a a 1

Note that is always equal to 1, since the prefix and the suffix of length are both equal to the word . Similarly, is 1 if and only if the first and the last letters are the same.


Autocorrelation polynomial

The autocorrelation polynomial of is defined as . It is a polynomial of degree at most .

For example, the autocorrelation polynomial of is and the autocorrelation polynomial of is . Finally, the autocorrelation polynomial of is .

Property

We now indicate some properties which can be computed using the autocorrelation polynomial.

First occurrence of a word in a random string

Suppose that you choose an infinite sequence of letters of , randomly, each letter with probability , where is the number of letters of . Let us call the expectation of the first occurrence of ? in ? . Then equals . That is, each subword of which is both a prefix and a suffix causes the average value of the first occurrence of to occur letters later. Here is the length of v.

For example, over the binary alphabet , the first occurrence of is at position while the average first occurrence of is at position . Intuitively, the fact that the first occurrence of is later than the first occurrence of can be explained in two ways:

  • We can consider, for each position , what are the requirement for 's first occurrence to be at .
    • The first occurrence of can be at position 1 in only one way in both case. If starts with . This has probability for both considered values of .
    • The first occurrence of is at position 2 if the prefix of of length 3 is or is . However, the first occurrence of is at position 2 if and only if the prefix of of length 3 is . (Note that the first occurrence of in is at position 1.).
    • In general, the number of prefixes of length such that the first occurrence of is at position is smaller for than for . This explain why, on average, the first arrive later than the first .
  • We can also consider the fact that the average number of occurrences of in a random string of length is . This number is independent of the autocorrelation polynomial. An occurrence of may overlap another occurrence in different ways. More precisely, each 1 in its autocorrelation vector correspond to a way for occurrence to overlap. Since many occurrences of can be packed together, using overlapping, but the average number of occurrences does not change, it follows that the distance between two non-overlapping occurrences is greater when the autocorrelation vector contains many 1's.

Ordinary generating functions

Autocorrelation polynomials allows to give simple equations for the ordinary generating functions (OGF) of many natural questions.

  • The OGF of the languages of words not containing is .
  • The OGF of the languages of words containing is .
  • The OGF of the languages of words containing a single occurrence of , at the end of the word is .

References

  • Flajolet and Sedgewick (2010). Analytic Combinatorics. New York: Cambridge University Press. pp. 60-61. ISBN 978-0-521-89806-5.
  • Rosen, Ned. "Expected waiting times for strings of coin flips" (PDF). Retrieved 3 December 2017.
  • Odlyzko, A. M.; Guibas, L. J. (1981). "String overlaps, pattern matching, and nontransitive games". Journal of Combinatorial Theory. Series A 30 (2): 183–208. doi:10.1016/0097-3165(81)90005-4.
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